In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
Blue Flibbins are so jealous of their red partners that they will not leave them on their own with any other bue Flibbin. What is the quickest way of getting the five pairs of Flibbins safely to. . . .
Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.
Baker, Cooper, Jones and Smith are four people whose occupations are teacher, welder, mechanic and programmer, but not necessarily in that order. What is each person’s occupation?
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
Three teams have each played two matches. The table gives the total number points and goals scored for and against each team. Fill in the table and find the scores in the three matches.
After some matches were played, most of the information in the table containing the results of the games was accidentally deleted. What was the score in each match played?
A huge wheel is rolling past your window. What do you see?
This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.
You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .
A game for 2 players that can be played online. Players take it in turns to select a word from the 9 words given. The aim is to select all the occurrences of the same letter.
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
Carry out cyclic permutations of nine digit numbers containing the digits from 1 to 9 (until you get back to the first number). Prove that whatever number you choose, they will add to the same total.
Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?
Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
Eight children enter the autumn cross-country race at school. How many possible ways could they come in at first, second and third places?
In the following sum the letters A, B, C, D, E and F stand for six distinct digits. Find all the ways of replacing the letters with digits so that the arithmetic is correct.
Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.
Your partner chooses two beads and places them side by side behind a screen. What is the minimum number of guesses you would need to be sure of guessing the two beads and their positions?
What happens when you add three numbers together? Will your answer be odd or even? How do you know?
Use your logical reasoning to work out how many cows and how many sheep there are in each field.
You have been given nine weights, one of which is slightly heavier than the rest. Can you work out which weight is heavier in just two weighings of the balance?
A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . .
Are these statements relating to odd and even numbers always true, sometimes true or never true?
A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.
Are these statements always true, sometimes true or never true?
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?
This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.
If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?
Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.
Look at three 'next door neighbours' amongst the counting numbers. Add them together. What do you notice?
Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
What does logic mean to us and is that different to mathematical logic? We will explore these questions in this article.
Can you find all the 4-ball shuffles?
Can you arrange the numbers 1 to 17 in a row so that each adjacent pair adds up to a square number?
Who said that adding couldn't be fun?
How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?
Here are some examples of 'cons', and see if you can figure out where the trick is.
If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.
Semicircles are drawn on the sides of a rectangle ABCD. A circle passing through points ABCD carves out four crescent-shaped regions. Prove that the sum of the areas of the four crescents is equal to. . . .
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.
Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.
Which hexagons tessellate?
Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?
In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?
Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?
Here are three 'tricks' to amaze your friends. But the really clever trick is explaining to them why these 'tricks' are maths not magic. Like all good magicians, you should practice by trying. . . .